The excess Gibbs energy of a certain binary mixture is equal to \(g R T x(1-\) \(x\) ) where \(\mathrm{g}\) is a constant and \(\mathrm{x}\) is the mole fraction of a solute \(\mathrm{A}\). Find an expression for the chemical potential of \(\mathrm{A}\) in the mixture and sketch its dependence on the composition.

In the context of thermodynamics and chemistry, chemical potential, symbolized as \( \mu \), is a fundamental concept that represents the change in the Gibbs energy (or free energy) of a system when an additional amount of a substance is introduced, keeping the temperature and pressure constant. The chemical potential of a component in a mixture is especially crucial as it dictates how substances will mix, react, or separate.

In practice, to obtain the chemical potential of a component \( A \) in a binary mixture, one starts with the excess Gibbs energy formula, \( G^{E} \), which in our exercise is given as \( G^{E} = gRTx(1-x) \). The excess component signifies how the real mixture deviates from ideality. The chemical potential of \( A \) is the partial molar Gibbs energy and can be calculated as the partial derivative of \( G^{E} \) with respect to the mole fraction of \( A \) (denoted as \( x \) in our case).

When looking at the chemical potential across different compositions within a mixture, we notice it provides insights about the stability and phase behavior of the mixture. In essence, the chemical potential is a driving force for mass transfer, phase changes, and chemical reactions. Understanding this concept is fundamental for students of chemistry and chemical engineering as it applies to processes like distillation, extraction, crystallization, and much more.

The partial derivative is an operation in calculus that shows how a function changes as one of its variables is varied while keeping other variables constant. In terms of chemical thermodynamics, the partial derivative is used to analyze the change in a property, such as Gibbs energy, with respect to changes in composition, temperature, or pressure, individually.

Specific to our exercise, the partial derivative \( \frac{\partial (G^{E} / n)}{\partial x} \) is calculated to find the chemical potential of component \( A \) in the mixture, denoted as \( \mu_A \), by differentiating the excess Gibbs energy equation with respect to the mole fraction, \( x \) of \( A \), and doing so while maintaining constant temperature and pressure. Thus, the partial derivative is a powerful tool in measuring the responsiveness of the chemical potential to changes in composition and can be visualized graphically to show the behavior of mixtures over a range of compositions.

The mole fraction is a way of expressing the concentration of a component in a mixture. It is defined as the ratio of the number of moles of that component to the total number of moles of all components in the mixture. Symbolized as \( x \) for a component \( A \), the mole fraction is dimensionless and varies between 0 and 1.

In the context of our exercise, \( x \) represents the mole fraction of solute \( A \) in a binary mixture. It has unique attributes such as being independent of temperature and pressure, and its sum with the mole fraction of other components in the mixture is always equal to 1. A key aspect of the mole fraction's utility is its role in determining the properties of mixtures, such as partial pressures in gases and vapor-liquid equilibrium in solutions.

Understanding mole fraction is essential when calculating the chemical potential because the composition-related variations in the chemical potential, as described by the given excess Gibbs energy function, directly depend on the mole fraction of the components. This relationship highlights the mole fraction's significance in the study of solution behavior and chemical thermodynamics, impacting real-world applications like formulating pharmaceuticals, designing separation processes, and analyzing biochemical reactions.